Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme32.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdleme32.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdleme32.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdleme32.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdleme32.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdleme32.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdleme32.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
8 |
|
cdleme32.c |
⊢ 𝐶 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) ) |
9 |
|
cdleme32.d |
⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
10 |
|
cdleme32.e |
⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
11 |
|
cdleme32.i |
⊢ 𝐼 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐸 ) ) |
12 |
|
cdleme32.n |
⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 ) |
13 |
|
cdleme32.o |
⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) |
14 |
|
cdleme32.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , 𝑂 , 𝑥 ) ) |
15 |
|
simp23l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑃 ≠ 𝑄 ) |
16 |
15
|
pm2.24d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( ¬ 𝑃 ≠ 𝑄 → 𝑋 ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) ) |
17 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝐾 ∈ HL ) |
18 |
17
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝐾 ∈ Lat ) |
19 |
|
simp21l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑋 ∈ 𝐵 ) |
20 |
|
simp11r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑊 ∈ 𝐻 ) |
21 |
1 6
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
22 |
20 21
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑊 ∈ 𝐵 ) |
23 |
1 2 4
|
latleeqm1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑊 ↔ ( 𝑋 ∧ 𝑊 ) = 𝑋 ) ) |
24 |
18 19 22 23
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑋 ≤ 𝑊 ↔ ( 𝑋 ∧ 𝑊 ) = 𝑋 ) ) |
25 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
26 |
18 19 22 25
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
27 |
|
simp21r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑌 ∈ 𝐵 ) |
28 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
29 |
18 27 22 28
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
30 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
31 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
32 |
|
simp13 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
33 |
|
simp31 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) |
34 |
1 2 3 4 5 6 7 8 9 10 11 12
|
cdleme27cl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑁 ∈ 𝐵 ) |
35 |
30 31 32 33 15 34
|
syl122anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑁 ∈ 𝐵 ) |
36 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑁 ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) → ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ∈ 𝐵 ) |
37 |
18 35 29 36
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ∈ 𝐵 ) |
38 |
|
simp33 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑋 ≤ 𝑌 ) |
39 |
1 2 4
|
latmlem1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) |
40 |
18 19 27 22 39
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) |
41 |
38 40
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) |
42 |
1 2 3
|
latlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑁 ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
43 |
18 35 29 42
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑌 ∧ 𝑊 ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
44 |
1 2 18 26 29 37 41 43
|
lattrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
45 |
|
breq1 |
⊢ ( ( 𝑋 ∧ 𝑊 ) = 𝑋 → ( ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ↔ 𝑋 ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) ) |
46 |
44 45
|
syl5ibcom |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( ( 𝑋 ∧ 𝑊 ) = 𝑋 → 𝑋 ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) ) |
47 |
24 46
|
sylbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑋 ≤ 𝑊 → 𝑋 ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) ) |
48 |
|
simp22 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) |
49 |
|
pm4.53 |
⊢ ( ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ↔ ( ¬ 𝑃 ≠ 𝑄 ∨ 𝑋 ≤ 𝑊 ) ) |
50 |
48 49
|
sylib |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( ¬ 𝑃 ≠ 𝑄 ∨ 𝑋 ≤ 𝑊 ) ) |
51 |
16 47 50
|
mpjaod |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑋 ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
52 |
14
|
cdleme31fv2 |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 ) |
53 |
19 48 52
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 ) |
54 |
|
simp1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) |
55 |
|
simp23 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) |
56 |
|
simp32 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) |
57 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
cdleme32a |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐹 ‘ 𝑌 ) = ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
58 |
54 27 55 33 56 57
|
syl122anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐹 ‘ 𝑌 ) = ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
59 |
51 53 58
|
3brtr4d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) |